Several people really liked my previous puzzle, including some high-school students 🙂 Some of you have sent me photos of your hand-written proofs. This level of enthusiasm for mathematics is totally awesome. Here is a detailed proof with some beautiful diagrams by Chris Grossack.
Here is my next puzzle:
Let and
and
.
Let and
and
.
Also,
All the above variables are positive integers and and
and
Prove or disprove the following claim:
Claim: There exists a rational number such that
and
are rational numbers.
If you want to take small baby steps towards a proof, start with the following special cases:
Special case 1: and
Special case 2: and
and
Quick homework problem: Prove the Special case 2 using a proof of my previous puzzle.
Have fun solving.
Some mathematician has said pleasure lies not in discovering truth, but in seeking it 🙂